<!DOCTYPE html>
<html lang="en-US">
<!--********************************************-->
<!--*       Generated from PreTeXt source      *-->
<!--*                                          *-->
<!--*         https://pretextbook.org          *-->
<!--*                                          *-->
<!--********************************************-->
<head>
<meta http-equiv="Content-Type" content="text/html; charset=UTF-8">
<meta name="robots" content="noindex, nofollow">
</head>
<body class="ignore-math">
<h3 class="heading"><span class="type">Paragraph</span></h3>
<p>Consider</p>
<div class="displaymath process-math" data-contains-math-knowls="./knowl/eq4_7.html ./knowl/eq4_10.html ./knowl/eq4_8.html ./knowl/eq4_9.html ./knowl/eq4_11.html ./knowl/eq4_12.html ./knowl/eq4_13.html ./knowl/eq4_14.html ./knowl/eq4_15.html ./knowl/eq4_16.html">
\begin{equation*}
L[y]=y^{(n)}+P_1(x) y^{(n-1)}+\cdots+P_{n-1}(x) y^{\prime}+P_n(x) y=g(x),
\end{equation*}
</div>
<p class="continuation">and</p>
<div class="displaymath process-math" data-contains-math-knowls="./knowl/eq4_7.html ./knowl/eq4_10.html ./knowl/eq4_8.html ./knowl/eq4_9.html ./knowl/eq4_11.html ./knowl/eq4_12.html ./knowl/eq4_13.html ./knowl/eq4_14.html ./knowl/eq4_15.html ./knowl/eq4_16.html">
\begin{equation}
L[y]=0.\tag{4.4.1}
\end{equation}
</div>
<p class="continuation">Suppose that we have found a fundamental set of solutions for (<a href="" class="xref" data-knowl="./knowl/eq4_7.html" title="Equation 4.4.1">(4.4.1)</a>), say <span class="process-math">\(y_1, y_2,\cdots, y_n\text{.}\)</span> Then the general solution is</p>
<div class="displaymath process-math" data-contains-math-knowls="./knowl/eq4_7.html ./knowl/eq4_10.html ./knowl/eq4_8.html ./knowl/eq4_9.html ./knowl/eq4_11.html ./knowl/eq4_12.html ./knowl/eq4_13.html ./knowl/eq4_14.html ./knowl/eq4_15.html ./knowl/eq4_16.html">
\begin{equation*}
y=C_1 y_1+C_2 y_2+\cdots+C_n y_n.
\end{equation*}
</div>
<p class="continuation">To find a particular solution, we suppose that it has the following form:</p>
<div class="displaymath process-math" data-contains-math-knowls="./knowl/eq4_7.html ./knowl/eq4_10.html ./knowl/eq4_8.html ./knowl/eq4_9.html ./knowl/eq4_11.html ./knowl/eq4_12.html ./knowl/eq4_13.html ./knowl/eq4_14.html ./knowl/eq4_15.html ./knowl/eq4_16.html">
\begin{equation}
Y=u_1(x) y_1+u_2(x) y_2+\cdots+u_n(x) y_n,\tag{4.4.2}
\end{equation}
</div>
<p class="continuation">where <span class="process-math">\(u_1, u_2, \cdots, u_n\)</span> are <span class="process-math">\(n\)</span> unknown functions (we can impose <span class="process-math">\(n-1\)</span> equations on <span class="process-math">\(u_1, u_2, \cdots, u_n\)</span> ourself). Then</p>
<div class="displaymath process-math" data-contains-math-knowls="./knowl/eq4_7.html ./knowl/eq4_10.html ./knowl/eq4_8.html ./knowl/eq4_9.html ./knowl/eq4_11.html ./knowl/eq4_12.html ./knowl/eq4_13.html ./knowl/eq4_14.html ./knowl/eq4_15.html ./knowl/eq4_16.html">
\begin{equation}
Y^{\prime}=u_1 y_1^{\prime}+u_2 y_2^{\prime}+\cdots u_n y_n^{\prime}+\underline{u_1^{\prime} y_1+u_2^{\prime} y_2+\cdots u_n^{\prime}y_n}.\tag{4.4.3}
\end{equation}
</div>
<p class="continuation">Impose that</p>
<div class="displaymath process-math" data-contains-math-knowls="./knowl/eq4_7.html ./knowl/eq4_10.html ./knowl/eq4_8.html ./knowl/eq4_9.html ./knowl/eq4_11.html ./knowl/eq4_12.html ./knowl/eq4_13.html ./knowl/eq4_14.html ./knowl/eq4_15.html ./knowl/eq4_16.html">
\begin{equation}
u_1^{\prime} y_1+u_2^{\prime} y_2+\cdots u_n^{\prime}y_n=0.\tag{4.4.4}
\end{equation}
</div>
<p class="continuation">Then</p>
<div class="displaymath process-math" data-contains-math-knowls="./knowl/eq4_7.html ./knowl/eq4_10.html ./knowl/eq4_8.html ./knowl/eq4_9.html ./knowl/eq4_11.html ./knowl/eq4_12.html ./knowl/eq4_13.html ./knowl/eq4_14.html ./knowl/eq4_15.html ./knowl/eq4_16.html">
\begin{equation}
Y^{\prime \prime}=u_1 y_1^{\prime \prime}+u_2 y_2^{\prime \prime}+\cdots+u_n y_n^{\prime \prime}+\underline{u_1^{\prime} y_1^{\prime}+u_2^{\prime} y_2^{\prime}+\cdots+u_n^{\prime} y_n^{\prime}}.\tag{4.4.5}
\end{equation}
</div>
<p class="continuation">Impose</p>
<div class="displaymath process-math" data-contains-math-knowls="./knowl/eq4_7.html ./knowl/eq4_10.html ./knowl/eq4_8.html ./knowl/eq4_9.html ./knowl/eq4_11.html ./knowl/eq4_12.html ./knowl/eq4_13.html ./knowl/eq4_14.html ./knowl/eq4_15.html ./knowl/eq4_16.html">
\begin{equation}
u_1^{\prime} y_1^{\prime}+u_2^{\prime} y_2^{\prime}+\cdots+u_n^{\prime} y_n^{\prime}=0.\tag{4.4.6}
\end{equation}
</div>
<p class="continuation">We repeat this process and finally have</p>
<div class="displaymath process-math" data-contains-math-knowls="./knowl/eq4_7.html ./knowl/eq4_10.html ./knowl/eq4_8.html ./knowl/eq4_9.html ./knowl/eq4_11.html ./knowl/eq4_12.html ./knowl/eq4_13.html ./knowl/eq4_14.html ./knowl/eq4_15.html ./knowl/eq4_16.html">
\begin{equation}
Y^{(n-1)}=u_1 y_1^{(n-1)}+u_2 y_2^{(n-1)}+\cdots+u_n y_n^{(n-1)}+\underline{u_1^{\prime} y_1^{(n-2)}+u_2^{\prime} y_2^{(n-2)}+\cdots+u_n^{\prime} y_n^{(n-2)} }.\tag{4.4.7}
\end{equation}
</div>
<p class="continuation">Impose that</p>
<div class="displaymath process-math" data-contains-math-knowls="./knowl/eq4_7.html ./knowl/eq4_10.html ./knowl/eq4_8.html ./knowl/eq4_9.html ./knowl/eq4_11.html ./knowl/eq4_12.html ./knowl/eq4_13.html ./knowl/eq4_14.html ./knowl/eq4_15.html ./knowl/eq4_16.html">
\begin{equation}
u_1^{\prime} y_1^{(n-2)}+u_2^{\prime} y_2^{(n-2)}+\cdots+u_n^{\prime} y_n^{(n-2)}=0.\tag{4.4.8}
\end{equation}
</div>
<p class="continuation">Substituting (<a href="" class="xref" data-knowl="./knowl/eq4_10.html" title="Equation 4.4.2">(4.4.2)</a>), (<a href="" class="xref" data-knowl="./knowl/eq4_8.html" title="Equation 4.4.3">(4.4.3)</a>), (<a href="" class="xref" data-knowl="./knowl/eq4_9.html" title="Equation 4.4.5">(4.4.5)</a>), (<a href="" class="xref" data-knowl="./knowl/eq4_11.html" title="Equation 4.4.7">(4.4.7)</a>) into the ODE,</p>
<div class="displaymath process-math" data-contains-math-knowls="./knowl/eq4_7.html ./knowl/eq4_10.html ./knowl/eq4_8.html ./knowl/eq4_9.html ./knowl/eq4_11.html ./knowl/eq4_12.html ./knowl/eq4_13.html ./knowl/eq4_14.html ./knowl/eq4_15.html ./knowl/eq4_16.html">
\begin{equation*}
\begin{aligned}
&amp;u_1^{\prime} y_1^{(n-1)}+u_2^{\prime} y_2^{(n-1)}+\cdots +u_n^{\prime} y_n^{(n-1)}\\
&amp;\qquad \quad+u_1 y_1^{(n)}+u_2 y_2^{(n)}+\cdots+u_n y_n^{(n)}\\
&amp;~+P_1(x) \left[ u_1 y_1^{(n-1)}+u_2 y_2^{(n-1)}+\cdots+u_n y_n^{(n-1)}     \right]\\
&amp;~+P_2(x) \left[  u_1 y_1^{(n-2)}+u_2 y_2^{(n-2)}+\cdots+u_n y_n^{(n-2)}    \right]\\
&amp;~+\cdots\\
&amp;~+P_n(x) \left[ u_1 y_1+u_2 y_2+\cdots+u_n y_n    \right]=g(x),
\end{aligned}
\end{equation*}
</div>
<p class="continuation">which is further arranged as</p>
<div class="displaymath process-math" data-contains-math-knowls="./knowl/eq4_7.html ./knowl/eq4_10.html ./knowl/eq4_8.html ./knowl/eq4_9.html ./knowl/eq4_11.html ./knowl/eq4_12.html ./knowl/eq4_13.html ./knowl/eq4_14.html ./knowl/eq4_15.html ./knowl/eq4_16.html">
\begin{equation*}
\begin{aligned}
&amp;u_1^{\prime} y_1^{(n-1)}+u_2^{\prime} y_2^{(n-1)}+\cdots +u_n^{\prime} y_n^{(n-1)}\\
&amp;+u_1 \left[y_1^{(n)}+P_1 y_1^{(n-1)}+\cdots+P_n y_1\right] \\
&amp;+u_2 \left[y_2^{(n)}+P_1 y_2^{(n-1)}+\cdots+P_n y_2\right]\\
&amp;\cdots\\
&amp;+u_n \left[y_n^{(n)}+P_1 y_n^{(n-1)}+\cdots+P_n y_n\right]=g(x).
\end{aligned}
\end{equation*}
</div>
<p class="continuation">The expressions inside the brackets are zero, so finally we have</p>
<div class="displaymath process-math" data-contains-math-knowls="./knowl/eq4_7.html ./knowl/eq4_10.html ./knowl/eq4_8.html ./knowl/eq4_9.html ./knowl/eq4_11.html ./knowl/eq4_12.html ./knowl/eq4_13.html ./knowl/eq4_14.html ./knowl/eq4_15.html ./knowl/eq4_16.html">
\begin{equation}
u_1^{\prime} y_1^{(n-1)}+u_2^{\prime} y_2^{(n-1)}+\cdots +u_n^{\prime} y_n^{(n-1)}=g(x).\tag{4.4.9}
\end{equation}
</div>
<p class="continuation">Up till now, we have obtained <span class="process-math">\(n\)</span> linear algebraic equations for <span class="process-math">\(u_1^{\prime}, u_2^{\prime},\cdots, u_n^{\prime}\text{,}\)</span> see (<a href="" class="xref" data-knowl="./knowl/eq4_12.html" title="Equation 4.4.4">(4.4.4)</a>), (<a href="" class="xref" data-knowl="./knowl/eq4_13.html" title="Equation 4.4.6">(4.4.6)</a>), (<a href="" class="xref" data-knowl="./knowl/eq4_14.html" title="Equation 4.4.8">(4.4.8)</a>), (<a href="" class="xref" data-knowl="./knowl/eq4_15.html" title="Equation 4.4.9">(4.4.9)</a>). They can be solved by using Cramer’s rule:</p>
<div class="displaymath process-math" data-contains-math-knowls="./knowl/eq4_7.html ./knowl/eq4_10.html ./knowl/eq4_8.html ./knowl/eq4_9.html ./knowl/eq4_11.html ./knowl/eq4_12.html ./knowl/eq4_13.html ./knowl/eq4_14.html ./knowl/eq4_15.html ./knowl/eq4_16.html">
\begin{equation}
u_1^{\prime}=\frac{g(x) W_1}{W(y_1, y_2, \cdots, y_n)},\cdots, u_m^{\prime}=\frac{g(x) W_m}{W(y_1, y_2, \cdots, y_n)},\cdots, u_n^{\prime}=\frac{g(x) W_n}{W(y_1, y_2, \cdots, y_n)},\tag{4.4.10}
\end{equation}
</div>
<p class="continuation">Note <span class="process-math">\(W(y_1, y_2, \cdots, y_n)\)</span> is the Wronskian as follows</p>
<div class="displaymath process-math" data-contains-math-knowls="./knowl/eq4_7.html ./knowl/eq4_10.html ./knowl/eq4_8.html ./knowl/eq4_9.html ./knowl/eq4_11.html ./knowl/eq4_12.html ./knowl/eq4_13.html ./knowl/eq4_14.html ./knowl/eq4_15.html ./knowl/eq4_16.html">
\begin{equation*}
W(y_1, y_2, \cdots, y_n)=\left| 
\begin{array}{ccccc}
y_1 &amp; \cdots &amp; y_m &amp; \cdots &amp; y_n\\
y_1^{\prime} &amp; \cdots &amp; y_m^{\prime} &amp; \cdots &amp; y_n^{\prime}\\
\vdots &amp; \quad &amp; \vdots &amp; \quad &amp; \vdots\\
y_1^{(n-1)} &amp; \cdots &amp; y_m^{(n-1)} &amp; \cdots &amp; y_n^{(n-1)}
\end{array}
\right|.
\end{equation*}
</div>
<p class="continuation">And <span class="process-math">\(W_m\)</span> is obtained by replacing the <span class="process-math">\(m\)</span>-th column of <span class="process-math">\(W(y_1, y_2, \cdots, y_n)\)</span> with <span class="process-math">\((0, 0, \cdots, 1)\text{:}\)</span></p>
<div class="displaymath process-math" data-contains-math-knowls="./knowl/eq4_7.html ./knowl/eq4_10.html ./knowl/eq4_8.html ./knowl/eq4_9.html ./knowl/eq4_11.html ./knowl/eq4_12.html ./knowl/eq4_13.html ./knowl/eq4_14.html ./knowl/eq4_15.html ./knowl/eq4_16.html">
\begin{equation*}
W_m=\left| 
\begin{array}{ccccc}
y_1 &amp; \cdots &amp; 0 &amp; \cdots &amp; y_n\\
y_1^{\prime} &amp; \cdots &amp; 0 &amp; \cdots &amp; y_n^{\prime}\\
\vdots &amp; \quad &amp; \vdots &amp; \quad &amp; \vdots\\
y_1^{(n-1)} &amp; \cdots &amp; 1&amp; \cdots &amp; y_n^{(n-1)}
\end{array}
\right|,\quad m=1, 2,\cdots, n.
\end{equation*}
</div>
<p class="continuation">Taking an integration to (<a href="" class="xref" data-knowl="./knowl/eq4_16.html" title="Equation 4.4.10">(4.4.10)</a>), one has</p>
<div class="displaymath process-math" data-contains-math-knowls="./knowl/eq4_7.html ./knowl/eq4_10.html ./knowl/eq4_8.html ./knowl/eq4_9.html ./knowl/eq4_11.html ./knowl/eq4_12.html ./knowl/eq4_13.html ./knowl/eq4_14.html ./knowl/eq4_15.html ./knowl/eq4_16.html">
\begin{equation*}
u_m=\int \frac{g(x) W_m}{W(y_1, y_2, \cdots, y_n)}\textrm{d} x+D_m,\quad m=1, 2, \cdots, n,
\end{equation*}
</div>
<p class="continuation">where <span class="process-math">\(D_m\)</span> is the arbitrary constant and we take it to be zero. Therefore, the particular solution is</p>
<div class="displaymath process-math" data-contains-math-knowls="./knowl/eq4_7.html ./knowl/eq4_10.html ./knowl/eq4_8.html ./knowl/eq4_9.html ./knowl/eq4_11.html ./knowl/eq4_12.html ./knowl/eq4_13.html ./knowl/eq4_14.html ./knowl/eq4_15.html ./knowl/eq4_16.html">
\begin{equation*}
Y=u_1 y_1+u_2 y_2+\cdots+u_n y_n=\sum_{m=1}^n u_m y_m=\sum_{m=1}^n y_m \int \frac{g(x) W_m}{W(y_1, y_2, \cdots, y_n)} \textrm{d} x.
\end{equation*}
</div>
<span class="incontext"><a href="sec4_4.html#p-168" class="internal">in-context</a></span>
</body>
</html>
